Multiplicative gerbes and Lie groups

Multiplicative gerbes are higher-categorical, geometric structures of lie groups that are compatible with the group structure. They correspond to higher-categorical generalizations of Lie group extensions, and realize in a geometrical way the cohomology of the classifying spaces of Lie groups in degree 3.

Multiplicative gerbes are in close relation to the classical theory of Lie groups and Lie algebras. Over compact, simple Lie groups all multiplicative gerbes with structure 2-group BU(1) can be constructed explicitly from Lie algebraic data.

In my research I develop a theory of connections on multiplicative gerbes, and pursue essentially two applications. The first application is to Chern-Simons theories; these are 3-dimensional topological field theories which are of great importance in Mathematics and Physics. Multiplicative gerbes can help to understand Chern-Simons theories with very general gauge groups, in particular non-simply connected ones. In this context the gerbe generalizes the so-called "level" of the Chern-Simons theory. An important open question is, for instance, in which way Chern-Simons theory fits into the context of so-called functorial field theories. I investigate possibilities to obtain such a description using multiplicative gerbes.

The second application is about transgression to the loop group of the underlying Lie group. Under transgression, a multiplicative gerbe with connection and abelian structure 2-group becomes a central extension of the loop group. These central extensions, in particular their representation theory, have been studied intensively within the last two decades. Multiplicative gerbes provide a finite-dimensional, higher-categorical perspective, alternatively to the classical, infinite-dimensional theory.

Below are some articles and manuscripts of talks about this topic.

Articles associated with this topic

  • Lie 2-groups from loop group extensions
  • Pontrjagin duality on multiplicative Gerbes
    together with Jaider Blanco, Bernardo Uribe
    J. Noncommut. Geom., to appear
  • Transgressive loop group extensions
    Math. Z. 286(1) 325-360, 2017
  • A Construction of String 2-Group Models using a Transgression-Regression Technique
    Analysis, Geometry and Quantum Field Theory, edited by C. L. Aldana, M. Braverman, B. Iochum, and C. Neira-Jiménez, volume 584 of Contemp. Math., pages 99-115, AMS, 2012
  • Lifting Problems and Transgression for Non-Abelian Gerbes
    together with Thomas Nikolaus
    Adv. Math. 242 (2013) 50-79
  • Polyakov-Wiegmann Formula and Multiplicative Gerbes
    together with Krzysztof Gawedzki
    J. High Energy Phys. 09 (2009) 073
  • Multiplicative Bundle Gerbes with Connection
    Differential Geom. Appl. 28(3), 313-340 (2010)

Talks associated with this topic

  • Multiplicative Gerbes and Chern-Simons Theory
    Seminar "Topologie", Universität Bonn, October 2009
  • Chern-Simons Theory and the Categorified Group Ring
    Workshop "Twisted K-Theory and Loop Groups", Breckenridge, Colorado, May 2010
  • Lectures on gerbes, loop spaces, and Chern-Simons theory
    Workshop "Chern-Simons Theory: Geometry, Topology and Physics", University of Pittsburgh, May 2013
  • Transgressive central extensions of loop groups
    Conference "Colloquium on Algebras and Representations - Quantum 2016", Universidad Nacional de Córdoba, March 2016
  • The stringor bundle
    Conference "Geometries from Strings and Fields", Galileo Galilei Institute, July 2023